Python implementation of Krotov's method for quantum optimal control

krotov, optimal-control, quantum-computing, quantum-mechanics, qutip
pip install krotov==1.2.1


Krotov Python Package

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Python implementation of Krotov's method for quantum optimal control.

This implementation follows the original implementation in the QDYN Fortran library. The method is described in detail in D. M. Reich, M. Ndong, and C. P. Koch, J. Chem. Phys. 136, 104103 (2012) (arXiv:1008.5126).

The krotov package is built on top of QuTiP.

Development happens on Github. You can read the full documentation at ReadTheDocs.

If you use the krotov package in your research, please cite it.


Optimal control is a cornerstones of quantum technology: relying not just on a passive understanding of quantum mechanics, but on the active utilization of the quantum properties of matter. Quantum optimal control asks how to manipulate the dynamics of a quantum system to behave in some desired way. This is essential for the realization of quantum computers, and related technologies such as quantum sensing. See e.g. Glaser et al. Eur. Phys. J. D 69, 279 (2015) for an overview of methods, applications, and current research directions. Quantum technology and thus quantum control are the focus of several large-scale national and super-national research endeavors, such as the U.S. $1 billion National Quantum Initiative Act and the €1 billion European Quantum Flagship program.

Krotov's method is one of the two leading gradient-based optimization algorithms used in numerical quantum optimal control. It simulates the dynamics of a quantum system under a set of initial controls, and evaluates the result with respect to an optimization functional to be minimized. It then iteratively modifies the controls to guarantee a monotonically decreasing value in the optimization functional. To date, there has not been an open source implementation of the method. This package provides that missing implementation.

The choice of Python as an implementation language is due to Python's easy to learn syntax, expressiveness, and immense popularity in the scientific community. Moreover, the QuTiP library exists to provide the foundations of numerically describing quantum systems, and already includes basic versions of some of the other popular algorithms in quantum control, the gradient-based GRAPE and the gradient-free CRAB. The availability of the Jupyter notebook framework provides an ideal platform for showing the use of the method.

The Krotov package targets both students wishing to enter the field of quantum control, and researchers in the field. By providing a comprehensive set of examples, we enable users of our package to explore the formulation of typical control problems, and to understand how Krotov's method can solve them. These examples are inspired by recent publications, and thus show the use of the method at the cutting edge of research. Optimal control is also increasingly important in the design of experiments, and we hope that the availability of an easy to use implementation of Krotov's will facilitate this further.

The use of Python implies that for large-scale control problems, performance may become a significant issue. In this case, it may be necessary to implement Krotov's method in a more efficient (compiled) language. While the method as such is relatively straightforward, there are some subtleties involved. Our implementation puts an emphasis on clarity, and the documentation provides detailed explanations of all necessary concepts. Thus, the Krotov package can serve as a reference implementation, leveraging Python's reputation as "executable pseudocode", and as a foundation against which to test other implementations.


The Krotov package is available for Python versions >= 3.5. Its main dependency is QuTiP (apart from the core packages of the Python scientific ecosystem). Thus, you should consider QuTiP's installation instructions.

In any case, using some sort of virtual environment is strongly encouraged. Most packages in the Python scientific ecosystem are now available as wheels, making installation via pip easy. However, QuTiP currently does not provide wheels. Thus, on systems that do not have the necessary compilers installed (Windows, macOS), the conda package manager provides a good solution.

Assuming conda is installed (e.g. through Miniconda), the following commands set up a virtual (conda) environment into which the Krotov package can then be installed:

$ conda create -n qucontrolenv python=3.6
$ conda activate qucontrolenv
$ conda config --append channels conda-forge
$ conda install qutip


To install the latest released version of krotov into your current (conda) environment, run this command in your terminal:

$ pip install krotov

This is the preferred method to install the krotov package, as it will always install the most recent stable release.

You may also do

$ pip install krotov[dev,extras]

to install additional development dependencies, including packages required to run the example notebooks.

If you don't have pip installed, the Python installation guide, respectively the Python Packaging User Guide can guide you through the process.

To install the latest development version of krotov from Github:

$ pip install git+


To use Krotov's method for quantum optimal control in a Python script or Jupyter notebook, start with:

import krotov


  • define the necessary quantum operators and states using QuTiP.
  • create a list of objectives, as instances of krotov.Objective
  • call krotov.optimize_pulses to perform an optimization of an arbitrary number of control fields over all the objectives.

See Using Krotov with QuTiP and Examples for details.